Contrasts in ANOVA. Decomposition of Treatment Sums of Squares using prior information on the structure of the treatments and/or treatment groups. Contrasts, notation….
Decomposition of Treatment Sums of Squares using prior information on the structure of the treatments and/or treatment groups
For a Oneway ANOVA, a contrast is a specific comparison of Treatment group means. Contrast constants are composed to test a specific hypothesis related to Treatment means based upon some prior information about the Treatment groups. For k treatment groups, contrast constants are a sequence of numbers
A given contrast will test a specific set of hypotheses:
to create an F-statistic with one numerator df.
Suppose we have two treatments which are supposed to be equivalent. For example, each of two drugs is supposed to work by binding to the receptor for adrenalin. Propanolol is such a drug sometimes used for hypertension or anxiety.
We may think that:
Control Drug A Drug B
Contrast 1 -1 ½ ½
Contrast 2 0 -1 +1
Contrast 1 tests whether or not the Control group differs from the groups which block the adrenalin receptors.
Contrast 2 tests whether or not the two drugs differ in their effect.
are orthogonal if:
The Trt Sums of Squares can be written as a sum of two Statistically independent terms:
Which can be used to test the hypotheses in the example. The a priori structure in the Treatments can be tested for significance in a more powerful way.
If all of the differences in the means are described by one of the contrasts, say the first contrast, then
is more likely to be significant than
Since the signal in the numerator is not combined with “noise”.
We can look into the Main Effects of Drug and Gender and Interaction via Orthogonal Contrasts
Drug A A B B
Gender M F M F
Contrast 1 +1/2 +1/2 -1/2 -1/2
Contrast 2 +1/2 -1/2 +1/2 -1/2
Contrast 3 +1/2 -1/2 -1/2 +1/2